\newproblem{lay:4_2_30}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.2.30}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $T:V\rightarrow W$ be a linear transformation from a vector space $V$ into a vector space $W$. Prove that the range of $T$ is a subspace of
	$W$. [Hint: typical elements of the range have the form $T(\mathbf{u})$ and $T(\mathbf{v})$ for $\mathbf{u},\mathbf{v}\in V$.]
}{
  % Solution
	We need to show that the range of $T$ meets the three requirements to be a subspace
	\begin{itemize}
		\item $\mathbf{0}_W\in \mathrm{Range}\{T\}$ \\
		      We know that for any linear transformation $T(\mathbf{0}_V)=\mathbf{0}_W$, so $\mathbf{0}_W$ is in the range of $T$.
		\item Given any two vectors $T(\mathbf{u}),T(\mathbf{v})\in \mathrm{Range}\{T\}$, $\in T(\mathbf{u})+T(\mathbf{v})\mathrm{Range}\{T\}$ \\
				  Since $T$ is a linear transformation
		      \begin{center}
						$T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$
					\end{center}
					So $T(\mathbf{u})+T(\mathbf{v})$ is also in the range of $T$.
		\item Given any vector $T(\mathbf{u})\in \mathrm{Range}\{T\}$ and $c\in \mathbb{R}$, $cT(\mathbf{u})\in \mathrm{Range}\{T\}$\\
		      Again, exploiting the fact that $T$ is linear
					\begin{center}
						$T(c\mathbf{u})=cT(\mathbf{u})$
					\end{center}
					So $cT(\mathbf{u})$ is in the range of $T$
	\end{itemize}
	Since $\mathrm{Range}\{T\}$ meets all properties, $\mathrm{Range}\{T\}$ is a subspace of $W$.
}
\useproblem{lay:4_2_30}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
